Connections Between Number Theory, Algebraic Geometry, and Combinatorics

数论、代数几何和组合数学之间的联系

• 批准号: 0901487
• 负责人: Matthew Baker
• 金额: $ 30.07万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901487&HistoricalAwards=false
• 关键词: Connections; Between; Number; Theory; Algebraic

This proposal is concerned with the development of new connections between arithmetic geometry, tropical geometry, Berkovich spaces, and combinatorics.The intellectual merit of the proposal lies primarily in the cross-fertilization between these different areas, and in the concrete applications being proposed. For example, the PI proposes to use ideas coming from algebraic geometry to provide new insight into the graph isomorphism problem, one of the most famous unsolved problems in graph theory and computer science.The PI will also show that harmonic morphisms, which play a prominent role in differential geometry and potential theory, arise naturally in arithmetic geometry, tropical geometry, and combinatorics. Applications will be given to a diverse array of subjects including component groups of Neron models, tropical intersection theory, and graph theory. Finally, the PI plans to develop new connections between Berkovich's theory of analytic spaces and tropical geometry. This will enable further development of the foundations of tropical geometry and potential theory on Berkovich spaces, and will also provide a more conceptual understanding of some recent results concerning tropical elliptic curves.The broader impacts of the proposed work will include applications to problems in the physical sciences, interaction with mathematicians in different fields, and support for undergraduate and graduate research. For example, the PI's new ideas on the graph isomorphism problem could potentially have applications to chemistry, biology, and computer science. Accomplishing the various goals laid out in this proposal will require the PI to interact with leading experts in the fields of number theory, algebraic geometry, combinatorics, and dynamical systems. The PI, who is currently supervising two graduate students and has been intensely involved for many years with undergraduate research, plans to work with students at all levels on research projects related to this proposal.

该提议涉及算术几何形状，热带几何形状，伯科维奇空间和组合学之间的新联系的发展。该提案的知识分子的优点主要在于这些不同领域之间的交叉施用以及提出的具体应用。 例如，PI建议使用来自代数几何形状的想法来提供对图形同构问题的新见解，这是图理论和计算机科学中最著名的未解决的问题之一。PI还将表明谐波形态，谐波形态起作用，它起着重要的作用在差异几何和潜在理论中的作用自然出现在算术几何形状，热带几何学和组合学中。 将应用于多种主题，包括Neron模型的组成组，热带交集理论和图理论。 最后，PI计划在伯科维奇的分析空间理论与热带几何形状之间建立新的联系。 这将进一步发展热带几何形状和对伯科维奇空间的潜在理论的基础，并且还将对有关热带椭圆曲线的一些最新结果提供更概念性的理解。拟议工作的更广泛影响将包括对物理中的问题的应用科学，与不同领域的数学家的互动以及对本科和研究生研究的支持。 例如，PI在图形同构问题上的新想法可能有可能在化学，生物学和计算机科学上应用。 完成本提案中规定的各种目标将要求PI与数字理论，代数几何，组合和动态系统的领先专家进行互动。 PI目前正在监督两名研究生，并且已经与本科生研究多年来一直参与其中，该PI计划与各级学生合作研究与该建议有关的研究项目。